Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions

Abstract

We elaborate on statistical thermodynamics models of relativistic geometric flows as generalizations of G. Perelman and R. Hamilton theory centred around C. Carathéodory axiomatic approach to thermodynamics with Pfaffian differential equations. The anholonomic frame deformation method, AFDM, for constructing generic off-diagonal and locally anisotropic cosmological solitonic solutions in the theory of relativistic geometric flows and general relativity is developed. We conclude that such solutions cannot be described in terms of the Hawking–Bekenstein thermodynamics for hypersurface, holographic, (anti-) de Sitter and similar configurations. The geometric thermodynamic values are defined and computed for nonholonomic Ricci flows, (modified) Einstein equations, and new classes of locally anisotropic cosmological solutions encoding solitonic hierarchies.

Appendices

Pfaffian differential equations

Let us provide a brief introduction into the theory of Pfaff forms and thermodynamics, see details and references in [37,38,39]. A Pfaff differential form is $\ \delta \phi =\sum \nolimits _{I}X_{I}{\text {d}}z^{I},$ where I runs integer values (for simplicity, we consider $I=1,2$) and $\delta f$ is differential 1-form but may be not a differential of a real valued function $\phi (z^{I})$ of real variables $z^{I},$ where $\partial _{I}:= \partial /\partial z^{I}$. An equation

$$\begin{aligned} \delta \phi =0 \end{aligned}$$

(A.1)

is called a non-exact Pfaff equation. If $\delta \phi ={\text {d}}\phi =(\partial _{I}\phi ){\text {d}}z^{I}$ is an exact differential of a function $\phi (z^{I}),$ i.e. we have an exact Pfaff equation, it is possible to integrate (A.1) along a path C connecting two points $z_{[1]}^{I}$ and $z_{[2]}^{I}$ (when $\phi$ is path-independent) and express the solution in the form

$$\begin{aligned} \phi =\int \nolimits _{C}{\text {d}}\phi =\phi (z_{[2]}^{I})-\phi (z_{[1]}^{I})=const. \end{aligned}$$

The H. A. Schwarz criterion is the necessary and sufficient condition to detect a total differential equation

$$\begin{aligned} \partial _{I}X_{J}=\partial _{J}X_{I}, \text{ for } I\ne J, \text{ i.e. } \frac{ \partial ^{2}\phi }{\partial x^{1}\partial x^{2}}=\frac{\partial ^{2}\phi }{ \partial x^{2}\partial x^{1}}. \end{aligned}$$

(A.2)

In many cases, a non-exact Pfaffian with $\partial _{I}X_{J}\ne \partial _{J}X_{I}$ can be transformed into an exact one by the aid of an integrating factor $K(z^{I}),$ when the coefficients of $\sum \nolimits _{I}KX_{I}{\text {d}}z^{I}$ satisfy the Schwarz condition

$$\begin{aligned} \partial _{I}(KX_{J})=\partial _{J}(KX_{I}), \text{ for } I\ne J. \end{aligned}$$

(A.3)

In such a case, the equation

$$\begin{aligned} K\delta \phi ={\text {d}}(K\phi )=0 \end{aligned}$$

(A.4)

can be integrated in an explicit form which allows us to find $\phi$ for any prescribed K satisfying (A.3).

In a more general context, if we are not able to transform (A.1) into a (A.4), we can additionally add to

$$\begin{aligned} \delta (K\phi )=\sum \limits _{I}KX_{I}{\text {d}}z^{I}\ne {\text {d}}(K\phi ) \end{aligned}$$

a differential of a new function $B(z^{I}),{\text {d}}B=(\partial _{I}B){\text {d}}z^{I}$ and search for such K and B when

$$\begin{aligned} \partial _{I}(KX_{J}+B)=\partial _{J}(KX_{I}+B), \text{ for } I\ne J \text{ and } \delta (K\phi )+{\text {d}}B={\text {d}}(K\phi +B). \end{aligned}$$

In such a case, we can integrate

$$\begin{aligned} {\text {d}}(K\phi +B)=0 \end{aligned}$$

(A.5)

for any suitable K and B and find $\phi$ in nonexplicit form from a so-called nonholonomic (non-integrable) function $F(\phi ,z^{I})=const.$ Usually, in thermodynamics we deal with equations of type (A.1) into a (A.4), but on nonholonomic manifolds, equations of type (A.5) are involved.

Parameterizations for families of cosmological d-metrics

We consider basic notations for quadratic line elements describing geometric flow evolutions and nonholonomic deformations of prime metrics into target cosmological ones.

1.1 Target d-metrics with geometric evolution of polarization functions

Families of target quadratic line elements can be represented in off-diagonal form, ${\mathbf {g}}_{\alpha \beta }=[g_{i},h_{a},n_{i},w_{i}],$ and/or using $\eta$-polarization functions

$$\begin{aligned} ds^{2}(\tau )= & {} g_{i}(\tau ,x^{k})[{\text {d}}x^{i}]^{2}+h_{3}(\tau ,x^{k},t)[{\text {d}}y^{3}+n_{i}(\tau ,x^{k},t){\text {d}}x^{i}]^{2}\nonumber \\&+h_{4}(\tau ,x^{k},t)[{\text {d}}t+w_{i}(\tau ,x^{k},t){\text {d}}x^{i}]^{2} \end{aligned}$$

(B.1)

$$\begin{aligned}= & {} \eta _{i}(\tau ,x^{k},t)\mathring{g}_{i}(x^{k},t)[{\text {d}}x^{i}]^{2}+\eta _{3}(\tau ,x^{k},t)\mathring{h}_{3}(x^{k},t)\nonumber \\&[{\text {d}}y^{3}+\eta _{i}^{3}(\tau ,x^{k},t)\mathring{N}_{i}^{3}(x^{k},t){\text {d}}x^{i}]^{2} \nonumber \\&+\eta _{4}(\tau ,x^{k},t)\mathring{h}_{4}(x^{k},t)[{\text {d}}t+\eta _{i}^{4}(\tau ,x^{k},t)\mathring{N}_{i}^{4}(x^{k},t){\text {d}}x^{i}]^{2} \nonumber \\= & {} \eta _{i}(\tau )\mathring{g}_{i}[{\text {d}}x^{i}]^{2}+\eta _{3}(\tau )\mathring{h} _{3}[{\text {d}}y^{3}+\eta _{k}^{3}(\tau )\mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2}\nonumber \\&+\eta _{4}(\tau )\mathring{h}_{4}[{\text {d}}t+\eta _{k}^{4}(\tau )\mathring{N} _{k}^{4}{\text {d}}x^{k}]^{2}, \end{aligned}$$

(B.2)

where $\tau$ is a temperature-like geometric evolution parameter and, for simplicity, we consider that prime metrics do not depend on such a parameter. There will be stated dependencies of type $\eta _{a}(\tau )=\eta _{a}(\tau ,x^{k},t)$ if such not notations do not result in ambiguities. We consider a coordinate transform to a new time-like coordinate $y^{4}=t\rightarrow \varsigma$ when $t=t(x^{i},\varsigma ),$

$$\begin{aligned} {\text {d}}t=\partial _{i}t{\text {d}}x^{i}+(\partial t/\partial \varsigma ){\text {d}}\varsigma ;{\text {d}}\varsigma =(\partial t/\partial \varsigma )^{-1}({\text {d}}t-\partial _{i}t{\text {d}}x^{i}), \text{ i.e. } (\partial t/\partial \varsigma ){\text {d}}\varsigma =({\text {d}}t-\partial _{i}t{\text {d}}x^{i}), \end{aligned}$$

and rewrite the target d-metric using the new time variable $\varsigma$ . For instance, the fourth term in (B.2) is computed

$$\begin{aligned}&\eta _{4}(\tau )\mathring{h}_{4}[{\text {d}}t+\eta _{k}^{4}(\tau )\mathring{N} _{k}^{4}{\text {d}}x^{k}]^{2}=\eta _{4}(\tau )\mathring{h}_{4}[\partial _{k}t{\text {d}}x^{k}+(\partial t/\partial \varsigma ){\text {d}}\varsigma +\eta _{k}^{4}(\tau ) \mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2} \\&\quad =\eta _{4}(\tau )\mathring{h}_{4}[(\partial _{k}t){\text {d}}x^{k}+(\partial t/\partial \varsigma ){\text {d}}\varsigma +\eta _{k}^{4}(\tau )\mathring{N} _{k}^{4}{\text {d}}x^{k}]^{2}\\&\quad =\eta _{4}(\tau )\mathring{h}_{4}[(\partial t/\partial \varsigma ){\text {d}}\varsigma +(\partial _{k}t+\eta _{k}^{4}(\tau )\mathring{N} _{k}^{4}){\text {d}}x^{k}]^{2} \\&\quad =\mathring{h}_{4}[\eta _{4}(\tau )(\partial t/\partial \varsigma ){\text {d}}\varsigma +\eta _{4}(\tau )(\partial _{k}t+\eta _{k}^{4}(\tau )\mathring{N} _{k}^{4}){\text {d}}x^{k}]^{2}\\&\quad =\mathring{h}_{4}[\eta _{4}(\tau )(\partial t/\partial \varsigma ){\text {d}}\varsigma +\eta _{4}(\tau )(\partial _{k}t/\mathring{N} _{k}^{4}+\eta _{k}^{4}(\tau ))\mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2} \end{aligned}$$

If $\eta _{4}\partial t/\partial \varsigma =1,$ when $\partial t/\partial \varsigma =(\eta _{4})^{-1}$ is introduced for ${\text {d}}t=\partial _{i}t{\text {d}}x^{i}+(\partial t/\partial \varsigma ){\text {d}}\varsigma ,$ we obtain

$$\begin{aligned} {\text {d}}t=(\partial _{i}t){\text {d}}x^{i}+(\eta _{4})^{-1}{\text {d}}\varsigma \text{ for } {\check{\eta }} _{k}^{4}=\eta _{4}(\partial _{k}t+\eta _{k}^{4}\mathring{N}_{k}^{4}). \end{aligned}$$

In result, a new time coordinate $\varsigma$ can be found from $\partial t/\partial \varsigma =(\eta _{4})^{-1}$ which results in

$$\begin{aligned} {\text {d}}\varsigma =\eta _{4}(x^{k},t){\text {d}}t;\varsigma =\int \eta _{4}(x^{k},t){\text {d}}t+\varsigma _{0}(x^{k}). \end{aligned}$$

Such coordinates with flow parameter $\tau$ and time-like $\varsigma$ are useful for computations of geometric evolution and nonholonomic deformations of the FLRW metrics.

1.2 Off-diagonal and diagonal parameterizations of prime d-metrics

Let us consider a target line quadratic element for an off-diagonal cosmological solution written in the form (B.2). We can introduce an effective target locally anisotropic cosmological scaling factor ${\check{a}} ^{2}(\tau ,x^{k},\varsigma ):= \eta (\tau ,x^{k},\varsigma )\mathring{a} ^{2}(x^{i},\varsigma )$ with gravitational polarization $\eta (\tau ,x^{k},\varsigma )$ and prime cosmological scaling factor $\mathring{a} ^{2}(\tau ,x^{i},\varsigma ),$ which allows to consider limits $\mathring{a} (\tau ,x^{i},\varsigma )\rightarrow \mathring{a}(\varsigma )$ with typical FLRW configurations. This can be performed following formulas

$$\begin{aligned} ds^{2}&=\eta _{3}(\tau )\{\frac{\eta _{i}(\tau )}{\eta _{3}(\tau )} \mathring{g}_{i}[{\text {d}}x^{i}]^{2}+\mathring{h}_{3}[{\text {d}}y^{3}+\eta _{k}^{3}\mathring{N }_{k}^{3}{\text {d}}x^{k}]^{2}\}+\mathring{h}_{4}[{\text {d}}\tau +{\check{\eta }}_{k}^{4}\mathring{ N}_{k}^{4}{\text {d}}x^{k}]^{2} \nonumber \\&={\check{a}}^{2}(\tau ,x^{k},\varsigma )\{{\check{\eta }}_{i}(\tau ,x^{k},\varsigma )\mathring{g}_{i}[{\text {d}}x^{i}]^{2}+\mathring{h}_{3}[{\text {d}}y^{3}+ {\check{\eta }}_{k}^{3}(\tau ,x^{k},\varsigma )\mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2} \}\nonumber \\&+\mathring{h}_{4}[{\text {d}}\varsigma +{\check{\eta }}_{k}^{4}(\tau ,x^{k},\varsigma ) \mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2} \nonumber \\&=\eta (\tau ,x^{k},\varsigma )\mathring{a}^{2}(\tau ,x^{i},\varsigma )\{ {\check{\eta }}_{i}(\tau ,x^{k},\varsigma )\mathring{g}_{i}[{\text {d}}x^{i}]^{2}+ \mathring{h}_{3}[{\text {d}}y^{3}+{\check{\eta }}_{k}^{3}(\tau ,x^{k},\varsigma ) \mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2}\}\nonumber \\&+\mathring{h}_{4}[{\text {d}}\varsigma +{\check{\eta }} _{k}^{4}(\tau ,x^{k},\varsigma )\mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2}, \nonumber \\ \text{ where } {\check{a}}^{2}(\tau ,x^{k},\varsigma )&:= \eta _{3}(\tau ,x^{k},t(x^{i},\varsigma ))=\eta (\tau ,x^{k},t(x^{i},\varsigma ))\mathring{a }^{2}(x^{k},t(x^{i},\varsigma ))=\eta (\tau ,x^{k},\varsigma )\mathring{a} ^{2}(x^{i},\varsigma ); \nonumber \\ {\check{\eta }}_{i}(\tau ,x^{k},\varsigma )&:= \frac{\eta _{i}(\tau ,x^{k},t(x^{i},\varsigma ))}{\eta (\tau ,x^{k},t(x^{i},\varsigma ))};\ {\check{\eta }}_{k}^{3}(\tau ,x^{k},\varsigma ):= \eta _{k}^{3}(\tau ,x^{k},t(x^{i},\varsigma ));\nonumber \\ {\check{\eta }}_{k}^{4}(\tau ,x^{k},\varsigma )&:= \eta _{4}\{\tau ,\partial _{k}t(x^{i},\varsigma )[\mathring{N}_{k}^{4}(x^{i},t(x^{i},\varsigma ))]^{-1}+\eta _{k}^{4}(\tau ,x^{i},t(x^{i},\varsigma ))\}\mathring{N} _{k}^{4}(x^{i},t(x^{i},\varsigma )). \end{aligned}$$

(B.3)

Considering a prime d-metric as a flat FLRW metric written in local coordinates ${\overline{u}}=\{{\overline{u}}^{\alpha }(x^{i},y^{3},\varsigma )=({\overline{x}} ^{1}(x^{i},y^{3},\varsigma ),{\overline{x}}^{2}(x^{i},y^{3},\varsigma ), {\overline{y}}^{3}(x^{i},y^{3},\varsigma ),{\overline{y}}^{4}(x^{i},y^{3}, \varsigma ))\},$ a d-metric (B.1) can be written in curved coordinate form $\mathring{a}^{2}({\overline{u}}),$ with local coordinated ${\overline{u}}^{\alpha }$ using a prime cosmological scaling factor $\mathring{ a}^{2}(\varsigma ),$

$$\begin{aligned} d\mathring{s}^{2}= & {} \mathring{a}^{2}({\overline{u}})\{\mathring{g}_{i}( {\overline{u}})[d{\overline{x}}^{i}]^{2}+\mathring{h}_{3}({\overline{u}})[d {\overline{y}}^{3}+\mathring{N}_{k}^{3}({\overline{u}})d{\overline{x}}^{k}]^{2}\}+ \mathring{h}_{4}({\overline{u}})[d{\overline{y}}^{4}+\mathring{N}_{k}^{4}( {\overline{u}})d{\overline{x}}^{k}]^{2}\rightarrow \mathring{a}^{2}(\varsigma )[{\text {d}}x^{{\check{i}}}]^{2}-{\text {d}}\varsigma ^{2}, \\ \text{ for } {\overline{u}}^{\alpha }\rightarrow & {} (x^{i},y^{3},\varsigma ), \mathring{g}_{i}\rightarrow 1,\mathring{h}_{3}\rightarrow 1,\mathring{h} _{4}\rightarrow -1,\mathring{N}_{k}^{a}({\overline{u}})\rightarrow 0 \text{ and } \mathring{a}^{2}({\overline{u}})\rightarrow \mathring{a}^{2}(\varsigma ). \end{aligned}$$

By definition, a quasi-FLRW configuration is stated by a diagonalized solution for a d-metric of type (B.3) when the integration functions and coordinates result in ${\check{\eta }}_{k}^{a}(\tau ,x^{k},\varsigma )=0,$

$$\begin{aligned} ds^{2}=\eta (\tau ,x^{k},\varsigma )\mathring{a}^{2}\{{\check{\eta }}_{i}(\tau ,x^{k},\varsigma )\mathring{g}_{i}[{\text {d}}x^{i}]^{2}+\mathring{h} _{3}[{\text {d}}y^{3}]^{2}\}+\mathring{h}_{4}[{\text {d}}\varsigma ]^{2}. \end{aligned}$$

(B.4)

Small nonholonomic deformations of such d-metrics can be parameterized ${\check{\eta }}_{i}\tau )\simeq$ $1+\varepsilon {\check{\chi }}_{i}(\tau ,x^{k},\varsigma )$ (see below formulas relevant to (B.9)) by the polarization of the target cosmological factor, $\eta (\tau ,x^{k},\varsigma )$ can be arbitrary one and not a value of $1+\varepsilon \chi (\tau ,x^{k},\varsigma )$ with a small parameter $\varepsilon .$ We can consider a resulting scaling factor $a^{2}(\tau ,x^{k},\varsigma )=\eta (\tau ,x^{k},\varsigma )\mathring{a}^{2}(x^{k},\varsigma ),$ with possible further re-parametrizations or limits to $a^{2}(\tau ,\varsigma )=\eta (\tau ,\varsigma )\mathring{a}^{2}(\varsigma )$ encoding possible nonlinear off-diagonal and parametric interactions determined by systems of nonlinear PDEs.

1.3 Approximations for flows of target d-metrics

To study nonlinear properties of cosmological models is convenient to consider different types of parameterizations and approximations for nonholonomic deformations of a prime metric to a target d-metric (B.3) being under geometric flow evolution. For our purposes, there are important six classes of exact, or parametric, solutions which can be generated by a respective subclass of generating functions and/or generating sources and, for certain cases, making some diagonal approximations, or by introducing small $\varepsilon$-parameters.

• 1. We can chose mutual re-parametrization of generating functions $(\Psi ,\Upsilon )\iff (\Phi ,\Lambda =const)$ and integrating functions when the coefficients of a family of target d-metric $\widehat{{\mathbf {g}}}_{\alpha \beta }(\tau ,\varsigma )$ depend only a time-like coordinate $\varsigma ,$ when $\eta (\tau ,x^{k},\varsigma )\rightarrow$ ${\widetilde{\eta }}(\tau ,\varsigma )$ and $a(\tau ,x^{k},\varsigma )\rightarrow {\widetilde{a}} ^{2}(\tau ,\varsigma )=$ ${\widetilde{\eta }}(\tau ,\varsigma )\mathring{a} ^{2}(\varsigma ).$ Respective families of linear quadratic elements (B.3) can be represented in the form

  $$\begin{aligned} ds^{2}(\tau )=\eta (\tau ,\varsigma )\mathring{a}^{2}(\varsigma )\{\check{ \eta }_{i}(\tau ,\varsigma )\mathring{g}_{i}[{\text {d}}x^{i}]^{2}+\mathring{h} _{3}[{\text {d}}y^{3}+{\check{\eta }}_{k}^{3}(\tau ,\varsigma )\mathring{N} _{k}^{3}{\text {d}}x^{k}]^{2}\}+\mathring{h}_{4}[{\text {d}}\varsigma +{\check{\eta }}_{k}^{4}(\tau ,\varsigma )\mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2}. \end{aligned}$$

  (B.5)

  With respect to coordinate bases, such families of cosmological solutions can be generic off-diagonal and could be chosen in some forms describing nonholonomic deformations of Bianchi cosmological models.

• 2. For FLRW prime configurations, we can consider families of generation functions and integration functions which result in zero values of the target N-connection coefficients under geometric flow evolutions and/or consider limits $\mathring{N}_{k}^{a}\rightarrow 0.$ For such cases, we can transform families (B.5) into families of diagonal metrics

  $$\begin{aligned} ds^{2}(\tau )=\eta (\tau ,\varsigma )\mathring{a}^{2}(\varsigma )\{\check{ \eta }_{i}(\tau ,\varsigma )\mathring{g}_{i}[{\text {d}}x^{i}]^{2}+\mathring{h} _{3}({\text {d}}y^{3})^{2}\}+\mathring{h}_{4}({\text {d}}\tau )^{2} \end{aligned}$$

  (B.6)

  modelling locally anisotropic interactions with a “memory” of nonholonomic/off-diagonal structures.

• 3. Flow evolution with small parametric nonholonomic deformations of a prime metric into families of target off-diagonal cosmological solutions (B.3) can be approximated

  $$\begin{aligned} {\check{\eta }}_{i}(\tau ,x^{k},\varsigma )\simeq 1+\varepsilon _{i}{\check{\chi }} _{i}(\tau ,x^{k},\varsigma ),\eta (\tau ,x^{k},\varsigma )\simeq 1+\varepsilon _{3}\chi (\tau ,x^{k},\varsigma ),{\check{\eta }}_{k}^{a}(\tau ,x^{k},\varsigma )\simeq 1+\varepsilon _{k}^{a}{\check{\chi }}_{k}^{a}(\tau ,x^{k},\varsigma ), \end{aligned}$$

  where small parameters $\varepsilon _{i},\varepsilon _{3},\varepsilon _{k}^{a}$ satisfy conditions of type $0\le |\varepsilon _{i}|,|\varepsilon _{3}|,|\varepsilon _{k}^{a}|\ll 1$ and, for instance, $\chi (\tau ,x^{k},\varsigma )$ is taken as a generating function. Such approximations restrict the class of generating functions subjected to nonlinear symmetries and may impose certain relations between such $\varepsilon$-constants and $\chi$-functions. Corresponding quadratic line elements can be parameterized

  $$\begin{aligned} ds^{2}(\tau )= & {} [1+\varepsilon _{3}\chi (\tau ,x^{k},\varsigma )]\mathring{a }^{2}(x^{i},\varsigma )\{[1+\varepsilon _{i}{\check{\chi }}_{i}(\tau ,x^{k},\varsigma )]\mathring{g}_{i}[{\text {d}}x^{i}]^{2} \nonumber \\&+\mathring{h}_{3}[{\text {d}}y^{3}+(1+\varepsilon _{k}^{3}{\check{\chi }}_{k}^{3}(\tau ,x^{k},\varsigma ))\mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2}\}+\mathring{h} _{4}[{\text {d}}\varsigma +(1+\varepsilon _{k}^{4}{\check{\chi }}_{k}^{4}(\tau ,x^{k},\varsigma ))\mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2}.\nonumber \\ \end{aligned}$$

  (B.7)

  Such $\tau$-families of off-diagonal solutions define cosmological metrics with certain small independent fluctuations, for instance, a FLRW embedded self-consistently into a locally anisotropic background under geometric flow evolution.

• 4. We can consider also families of off-diagonal cosmological solutions with small parameters $\varepsilon _{i},\varepsilon _{3},\varepsilon _{k}^{a}$ when the generating functions and d-metric and N-connection coefficients do not depend on space like coordinates, which is typical for a number of cosmological models. For such approximations, the family of quadratic line element (B.7) transforms into

  $$\begin{aligned} ds^{2}(\tau )= & {} [1+\varepsilon _{3}\chi (\tau ,\varsigma )]\mathring{a} ^{2}(\varsigma )\{[1+\varepsilon _{i}{\check{\chi }}_{i}(\tau ,\varsigma )] \mathring{g}_{i}[{\text {d}}x^{i}]^{2} \nonumber \\&+\mathring{h}_{3}[{\text {d}}y^{3}+(1+\varepsilon _{k}^{3}{\check{\chi }}_{k}^{3}(\tau ,\varsigma ))\mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2}\}+\mathring{h}_{4}[{\text {d}}\varsigma +(1+\varepsilon _{k}^{4}{\check{\chi }}_{k}^{4}(\tau ,\varsigma ))\mathring{N} _{k}^{4}{\text {d}}x^{k}]^{2}. \end{aligned}$$

  (B.8)

• 5. There are off-diagonal deformations, for instance, of a FLRW metric into a family of locally anisotropic cosmological solutions which can be constructed using only one small parameter $\varepsilon =\varepsilon _{i}=\varepsilon _{3}=\varepsilon _{k}^{a},$ and when the formulas (B.8) transform into

  $$\begin{aligned} ds^{2}(\tau )= & {} [1+\varepsilon \chi (\tau ,x^{k},\varsigma )]\mathring{a} ^{2}(x^{i},\varsigma )\{[1+\varepsilon {\check{\chi }}_{i}(\tau ,x^{k},\varsigma )]\mathring{g}_{i}[{\text {d}}x^{i}]^{2} \nonumber \\&+\mathring{h}_{3}[{\text {d}}y^{3}+(1+\varepsilon {\check{\chi }}_{k}^{3}(\tau ,x^{k},\varsigma ))\mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2}\}+\mathring{h} _{4}[{\text {d}}\varsigma +(1+\varepsilon {\check{\chi }}_{k}^{4}(\tau ,x^{k},\varsigma )) \mathring{N}_{k}^{4}{\text {d}}x^{k}]^{2}.\nonumber \\ \end{aligned}$$

  (B.9)

  Such flows with $\varepsilon$-deformations can be generated by corresponding small $\varepsilon$-deformations of flows generating functions.

• 6. We can impose on families (B.9) the condition that the $\varepsilon$-deformations depend only on evolution temperature like parameter and a time-like coordinate. This results in d-metrics

  $$\begin{aligned} ds^{2}(\tau )= & {} [1+\varepsilon \chi (\tau ,\varsigma )]\mathring{a} ^{2}(\varsigma )\{[1+\varepsilon {\check{\chi }}_{i}(\tau ,\varsigma )] \mathring{g}_{i}[{\text {d}}x^{i}]^{2} \\&+\mathring{h}_{3}[{\text {d}}y^{3}+(1+\varepsilon {\check{\chi }}_{k}^{3}(\tau ,\varsigma ))\mathring{N}_{k}^{3}{\text {d}}x^{k}]^{2}\}+\mathring{h}_{4}[{\text {d}}\varsigma +(1+\varepsilon {\check{\chi }}_{k}^{4}(\tau ,\varsigma ))\mathring{N} _{k}^{4}{\text {d}}x^{k}]^{2} \end{aligned}$$

  which can be considered as some ansatz used, for instance, for describing geometric evolution of quantum fluctuations of FLRW metrics.

In various classes of cosmological models with families of solutions with parametric $\varepsilon$-decompositions can be performed in a self-consistent form by omitting quadratic and higher order terms after a class of locally anisotropic solutions have been found for some general data $(\eta _{\alpha },\eta _{i}^{a}).$ They are more general than approximate solutions found, for instance, for classical and quantum fluctuations of standard FLRW metrics and may involve flow evolution parameters of cosmological constants and generating functions and sources. For certain subclasses of generic off-diagonal solutions, we can consider that $\varepsilon _{i},\varepsilon _{a},\varepsilon _{i}^{a}\sim \varepsilon ,$ when only one small parameter is considered for all coefficients of nonholonomic deformations.